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Notes on the Calderón Problem with Partial Data
| Content Provider | Semantic Scholar |
|---|---|
| Author | Valkenburgh, Michael Van |
| Copyright Year | 2009 |
| Abstract | These notes are my commentary on the paper “The Calderón Problem with Partial Data” by Kenig, Sjöstrand, and Uhlmann. 1. One of Calderón’s Main Characteristics In the following pages, I give a nearly line-by-line discussion of the paper “The Calderón Problem with Partial Data” by Kenig, Sjöstrand, and Uhlmann [14]. But before I begin, it is best to reflect on Alberto P. Calderón himself, and the way he worked. We recall that “one of Calderón’s main characteristics [is that] he always sought his own proofs, developed his own methods. From the start, Calderón worked in mathematics that way: he rarely read the work of others farther than the statements of theorems, and after grasping the general nature of the problem, went ahead by himself. In this process, Calderón not only rediscovered results, but added new insights to the subject” [4]. For those who wish to consider the problem themselves before reading about someone else’s methods, in this brief section I only state the main problem. Let Ω ⊂⊂ R be a bounded open connected set with, say, C∞ boundary. For q ∈ L∞(Ω) we consider the operator −∆ + q : L(Ω) → L(Ω) with domain H(Ω) ∩ H 0 (Ω), and we assume that 0 is not an eigenvalue of −∆ + q : H(Ω) ∩H1 0 (Ω) → L(Ω). Under this assumption, we have a well-defined Dirichlet-to-Neumann (DN) map Nq : H(∂Ω) 3 v 7→ ∂νu|∂Ω ∈ H−1/2(∂Ω), where ν denotes the exterior unit normal and u is the unique solution in H∆(Ω) := {u ∈ H(Ω); ∆u ∈ L2(Ω)} of the problem (−∆ + q)u = 0 in Ω, u|∂Ω = v. Question: Let q1, q2 be two functions as above. Given two subsets Γ1, Γ2 ⊂ ∂Ω, we would like to say that if Nq1u = Nq2u in Γ1, for all u ∈ H(∂Ω) ∩ E ′(Γ2), then q1 = q2. What are conditions on Γ1 and Γ2 such that this is true? ¥ Date: August 5, 2009. 1 2 MICHAEL VANVALKENBURGH |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.math.ucla.edu/~mvanvalk/NotesOnCalderonsProblem.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
